Related papers: On the maximum principle for parabolic equations w…

In this paper we obtain new estimates of the sequential Caputo fractional derivatives of a function at its extremum points. We derive comparison principles for the linear fractional differential equations, and apply these principles to…

We present a new, short proof of the increased regularity obtained by solutions to uniformly parabolic partial differential equations. Though this setting is fairly introductory, our new method of proof, which uses a priori estimates, can…

We prove the Lp,q-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the…

This paper presents a maximum principle-based approach in the establishment of input-to-state stability (ISS) for a class of nonlinear parabolic partial differential equations (PDEs) over higher dimensional domains with variable…

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the time discontinuous Galerkin solution of linear parabolic equations. Such estimates have many applications. They…

In this paper, we employ a space-time finite element method to discretize the parabolic initial-boundary value problem and extend its error analysis with refined estimates on unstructured space-time meshes. We establish higher-order…

This is a generalization of our prior work on the compact fixed point theory for the elliptic Rosseland-type equations. We obtain the maximum principle without the technical Steklov techniques. Inspired by the Rosseland equation in the…

This is an English translation of the following paper, published several years ago: Nikonorov Yu.G., Nikonorova Yu.V. Generalized Popoviciu's problem (Russian), Tr. Rubtsovsk. Ind. Inst., 7, 229-232 (2000), Zbl. 0958.51021. All inserted…

We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and…

In this article we first establish the maximum principle of the antisymmetric functions for parabolic fractional $p$-equations. Then we use it and the parabolic inequalities to provide a different proof of symmetry and monotonicity for…

Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…

In this work there is established an optimal existence and regularity theory for second order linear parabolic differential equations on a large class of noncompact Riemannian manifolds. Then it is shown that it provides a general unifying…

The goal of this note is to give the unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems studied by R.~Bryant and P.~Griffiths. In this note we list several classical…

The regularization of propagators by means of a complex metric is considered. (The paper is an English translation of the first of two articles in Russian published by the author in 1987-88: V.D. Ivashchuk, Regularization by…

We establish the unique solvability in weighted mixed-norm Sobolev spaces for a class of degenerate parabolic and elliptic equations in the upper half space. The operators are in nondivergence form, with the leading coefficients given by…

We give a short, simple proof of maximal regularity for linear parabolic evolution equations on manifolds with cylindrical ends by making use of pseudodifferential parametrices and the concept of R-boundedness for the resolvent.

Inspired by Ogawa-Shimizu [JEE 2022] and Chen-Liang-Tsai [IMRN 2025] on the second and first order derivative estimates of solution of heat equation in the upper half space with boundary data in homogeneous Besov spaces, we extend the…

We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our…

A class of (possibly) degenerate integro-differential equations of parabolic type is considered, which includes the Kolmogorov equations for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential…

We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in…